Bayesian Optimization of Bilevel Problems
Omer Ekmekcioglu, Nursen Aydin, Juergen Branke
TL;DR
The paper tackles the challenge of efficiently solving bilevel optimization problems where both the upper and lower level objectives are expensive black-box functions. It introduces BILBAO, a multi-task Bayesian optimization framework that models the upper- and lower-level functions as separate Gaussian processes over the joint decision space, enabling information transfer between levels via a learned lower-level response map $\Phi$. Central to the approach are REVI and REVITS acquisition strategies that focus sampling on informative regions of the lower-level map, guided by upper-level uncertainty through Thompson sampling. Empirical results on 2D and 4D synthetic benchmarks show that BILBAO achieves faster and more robust convergence than state-of-the-art benchmarks, highlighting improved sample efficiency and resilience to lower-level solution quality. These findings suggest a practical pathway for solving expensive hierarchical decision problems in economics, engineering, and machine learning with significant gains in efficiency.
Abstract
Bilevel optimization, a hierarchical mathematical framework where one optimization problem is nested within another, has emerged as a powerful tool for modeling complex decision-making processes in various fields such as economics, engineering, and machine learning. This paper focuses on bilevel optimization where both upper-level and lower-level functions are black boxes and expensive to evaluate. We propose a Bayesian Optimization framework that models the upper and lower-level functions as Gaussian processes over the combined space of upper and lower-level decisions, allowing us to exploit knowledge transfer between different sub-problems. Additionally, we propose a novel acquisition function for this model. Our experimental results demonstrate that the proposed algorithm is highly sample-efficient and outperforms existing methods in finding high-quality solutions.